International Journal of Computer Applications (0975 – 8887)
Volume 48– No.22, June 2012
Harmony search to solve the container storage problem
with different container types
I. Ayachi
R. Kammarti
M.Ksouri
P. Borne
LACS, ENIT, TunisBelvédère Tunisie
LACS, ENIT, TunisBelvédère Tunisie
LACS, ENIT, Tunis-Belvédère
Tunisie,
LAGIS, ECL, Villeneuve
d’Ascq, France
ABSTRACT
This paper presents an adaptation of the harmony search
algorithm to solve the storage allocation problem for inbound
and outbound containers. This problem is studied considering
multiple container type (regular, open side, open top, tank,
empty and refrigerated) which lets the situation more
complicated, as various storage constraints appeared. The
objective is to find an optimal container arrangement which
respects their departure dates, and minimize the re-handle
operations of containers.
The performance of the proposed approach is verified
comparing to the results generated by genetic algorithm and
LIFO algorithm.
General Terms
Container storage problem, metaheuristics.
Keywords
Harmony search, Genetic algorithm, transport scheduling,
metaheuristic, optimization, container storage
1. INTRODUCTION
The container storage space allocation is a critical decision in
container terminals. It influences the productivity of the
unloading process, either for inbound or outbound containers.
It’s a complex operation since it is highly inter-related with
the routing of yard crane and truck [17].
This paper focuses on optimizing the way of allocating
inbound and outbound containers in storage locations, known
as the storage space allocation problem (SSAP). This problem
is classified as a three dimensions bin-packing problem where
containers are the items and storage spaces in the port
represent the used bins. It falls into the category of NP hard
problems. Generally, this problem is studied considering a
single container type. However, this does not stand the
problem under its real-life statement as there are multiple
container types that should be considered, (refrigerated, open
side, empty, dry, open top and tank). This lets the problem
more complicated, as various constraints appeared, related to
the container type’s requirements (e.g. refrigerated containers
must be allocated to the blocks equipped by the power point,
on an open top container, we cannot place a container at the
top, tank container must be placed on each other, etc.)
Making a storage space allocation decision for different types
of containers is too complicated especially for large scale
instances and it is hard, even impossible, to solve it optimally.
Therefore, most of the proposed solution approaches are
based on metaheuristics.
A metaheuristic is a computational method seeking for a good
solution in a reasonable computation time without being able
to guaranty optimality. Some of these approaches are based on
the gradient method, which presents some limits such as the
fact that they are often trapped in a local optimal especially
for complex optimization problems having several local
optimums.
Due to this restriction, other metaheuristics are developed
based on simulation, to solve complex problems. They imitate
natural phenomena such as the genetic algorithm inspired by
biological evolutionary process [8], ant colony [5], the
harmony search [7], firefly algorithm [21], cuckoo search
[20].
There is a large number of metaheuristics and it is difficult to
find the appropriate one for a specific problem, especially in
the absence of benchmarks. One way to face this dilemma is
to use multiple approaches, compare them and select the one
generating the best result.
In this paper a Harmony Search (HS) algorithm is proposed to
solve the problem of storage space allocation of containers
with different types. To evaluate the performance of this
method, we compare his results with those generated by the
genetic algorithm described in [1] and the Last In First Out
algorithm.
Harmony search algorithm was proposed by [7]. It was
successfully applied to solve various engineering optimization
problems such as vehicle routing [6], reliability [23],
structural optimization [15] and function optimisation [18]
The rest of this paper is organized as follows: In section 2, a
literature review for the container storage problem is
presented. The mathematical formulation of the problem is
given, in section 3. Next in section
4, the Harmony Search algorithm is described. Section 5 is
devoted to the description of the Harmony search adaptation
to the SSAP. Then, some experiments and results are
presented and discussed, in section 6. Section 7 included a
comparative study of the proposed approach with the genetic
algorithm and the Last in First out (LIFO). Finally, section 8
covers our conclusion.
2. LITERATURE REVIEW
The container storage space allocation is the most difficult
task in container terminals since inbound and outbound
containers are stacked together in the same storage area. After
arrival at the terminal, each container picked up by
transportation equipment and affected to one of the storage
blocks. When the designated ship arrived, containers are
unloaded from yard block, transported to the berth and loaded
onto the vessel. The chain of operations for import containers
are performed in the reverse order [10].
The container storage space allocation problem (SSAP)
consists on affecting each container to the most suitable place
in the storage area. The containers are often arranged with the
objective of reducing the number of handling operations
required later on to load/unload containers.
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International Journal of Computer Applications (0975 – 8887)
Volume 48– No.22, June 2012
In the literature, various papers were proposed, treating
different variants of the problem. Some of them will be
presented in this section.
Kim and Park [11] proposed a heuristic decision rule and a
sub-gradient optimization technique to solve the storage space
allocation for outbound containers. Their objective was to find
an arrangement of the containers that exploits efficiently the
storage space and loading operations.
Preston and Kozan [19] proposed a genetic algorithm to solve
the container location model at seaport terminals. Their
objective was to reduce the transfer and the handling time of
containers. This approach took the Brisbane port as a case
study and generated good results in comparison to the process
already used in this port.
Kim [12] presented a technique to estimate the rehandlings
number for the next pick-up and the total number of rehandles
to pick up all inbound containers in a bay.
Kim and Kim [13] proposed a cost model to estimate various
cost components related to the import container handling and
to determine subsequently the storage space and the number
of transfer cranes required.
Also, in [14], a prediction model of unloading containers
times and equipment utilization is presented.
Chen and col. [4] combined diverse meta-heuristics (tabu
search, simulated annealing and genetic algorithms) to solve
the port yard storage optimization problem. It aims to
minimize the space allocated to the cargo within a time
interval.
Lee and col. [16] developed a heuristic algorithm to solve the
yard truck scheduling and the storage allocation problems.
Their objective is to minimize the weighted sum of total delay
of requests and the cost of total travel time of yard trucks.
Zhang and col. [22] solved the (SSAP) using a rolling-horizon
approach. Both outbound and inbound containers are
considered .Their aim was to minimize the total transportation
distance of containers between blocks and vessel berthing
locations.
In [2], a harmony search algorithm is proposed to solve the
SSAP where a single container type was considered. Its aim
was to reduce the re-handle operations of containers. The
results were compared to a genetic algorithm previously
applied to the same problem in [9] and recorded good results.
Bazzazi and al. [3] extended the SSAP proposed in the
literature [22], where different containers types and sizes are
considered simultaneously. The authors proposed a genetic
algorithm to solve this problem and they supposed that the
allowable blocks to which a container type can be allocated
are known in advance.
Ayachi and col. [1] developed a genetic algorithm to solve the
problem of allocating containers of multiple types, in storage
spaces in the port. The results generated by the proposed
approach were compared to a Last in First out (LIFO)
algorithm.
In this paper, a harmony search is applied to solve the SSAP
considering multiple containers types (refrigerated, open side,
empty, dry, open top and tank).
3. PROBLEM FORMULATION
In this section, we detail our evolutionary approach by
presenting the adopted mathematical formulation based on the
following assumptions.
3.1 Assumptions

Initially containers are unloaded from the vessel and
transmitted to storage area waiting for allocation in the
allowable places of the storage block.

To unload a container, all containers above must be rehandled.

Each container has departure time.

The initial state of storage blocks, available places, is
known and to be considered in the load planning.

The containers are of different types (dry, open top, open
side, tank, empty and refrigerated).

Containers have the same size
The storage area in the port is composed of several blocks
which can be equipped by a power point to store reefer
containers or regular blocks for the other container types.
Figure 1 shows an example of a storage area.
Fig 1. Storage area
3.2 Input parameters
Let’s consider the following variables:
 i : Container index, i = 1, …, Nc
 b : storage block index; b = 1, …, NBlock
 NBlock = Nstock_reg + Nstock_refrig : le nombre de blocks
disponibles
 Nstock_reg : the number of storage blocks for containers
don’t requiring a power point
 Nstock_refrig : the number of storage blocks for refrigerated
containers.
 Nc : the number of containers to stored.
 di : departure date of container i
 NcFloor (j,b) : the number containers in the floor j of the
block b
 n1 : Maximum containers number on the axis X
 n2 : Maximum containers number on the axis Y
 n3 : Maximum containers number on the axis Z
 NT : the number of container types

Nc(T) : the number of containers of type T,
where :
In this work we suppose that:
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International Journal of Computer Applications (0975 – 8887)
Volume 48– No.22, June 2012
1
2
3
T  4
5
6

Nc NT
 C j, r (x, y, z  1, b)  1
Ci, t (x, y, z, b) - 
j  1r  1
 i  [1..Nc],  t  3,4
if it's a dry container
if it's an empty container
if it's an open topcontainer
if it's an open side container
if it's a tank container
if it's a reefer container
(5)
The constraint 5 enssures that an open top container or an
 Ncmax: Maximum containers number, with Ncmax =
(Nstock_reg + Nstock_refrig ) n1.n2.n3
3.3 Decision Variable
For this problem, Ci,t(x, y, z, b) designates the decision
variable.
1 if thereis a containeri in theposition

(x,
y,
z,
b)

(1)
C i, t
 x, y , z at theblock b
0 otherwise

x  [1,.., n1], y  [1,.., n2], z  [1,.., n3]
open side container can not have another container above.
Nc NT n 3 - z
 C j, r (x  m, y, z, b)  1 (6)
Ci,4 (x, y, z, b) -  
j  1r  1 m  1
 i  [1..Nc]
The constraint 6 indicates that there aren’t any containers at
the open side of container type 4 (open side container
C
i,6
1 , Si the block is reefer
(x, y, z, b)  
0, Otherwise
(7)
3.4 Mathematical formulation
The constraint 7 suggests that a reefer container must be
The main objective of the studied problem is to optimize a
fitness function that aims to reduce the number of container
rehandlings and then minimize the ship stoppage time.
allocated to the blocks equipped by the power point.
This function can be described as follows:
NT Nc(T) N block
Min 

t 1 i 1

b 1
m i ,b (d i ) C i, t (x, y, z, b)
(2)
Where:
 Mi,b (di) : the minimum number of container rehandles to
unload the container i which is in the storage block b. Mi,b
is equal to the number of container above the container i,
in the same stack and having a departure time greater than
di
Fig 2. The extraction of container B
Nc
C i,5 (x, y, z, b) -  C j ,r (x, y, z  1, b)  1
j 1
 i  [1..Nc], r  1, 2, 3, 4, 6
(8)
The constraint 8 indicates that tank containers must be placed
on each other
4. HARMONY SEARCH
The harmony search algorithm is developed to imitate the
musician behavior.
HS is based on the analogy with the music improvisations
process seeking for the best harmony. The harmony in music
is analogous to the optimization solution vector, and the ideal
harmony is analogous to optimal solution. The musical
harmony is improved practice after practice using the set of
the pitches played by each instrument. Also, the fitness
function is improved iteration by iteration using the values
assigned for decision variables. Figure 3 shows this analogy.
HS does not require initial values for the decision variables.
Additionally, it uses a stochastic random search based on the
harmony memory considering rate and the pitch adjusting rate
so that derivative information is unnecessary.
Compared to earlier meta-heuristic optimization algorithms,
the HS algorithm imposes fewer mathematical requirements.
So, it can be easily adopted for various types of engineering
optimization problems [15]
3.5 Constraints
The model is subject to the following constraints:
Nc floor (j, b)  Nc floor ((j  1), b)
(3)
 j  [1..n 3 - 1],  b  [1..N block ]
Nc NT
C i, t (x, y, z, b) -   C j, r (x, y, z  1, b)  0
j  1r  1
 i  [1..Nc],  t  [1..NT ]
(4)
The constraint equations (3) and (4) ensure that a floor lower
level contains more containers than the one directly above.
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International Journal of Computer Applications (0975 – 8887)
Volume 48– No.22, June 2012
 x 11
 2
x 1

.
.

.
 x HMS-1
 1
 x HMS
 1
x 12
...
x 1N -1
x 1N
x 22
...
x 2N -1
x 2N
...
...
...
...
...
...
...
...
...
-1
x HMS
...
2
x HMS
...
2
...
...
...
-1
-1
x HMS
x HMS
N -1
N
x HMS
N -1
x HMS
N


f(x 2 ) 

...


...

...

f(x HMS -1 )

f(x HMS ) 
f(x 1 )
(9)
4.3 New harmony improvisation
Fig 3. Analogy between musical improvisations and
optimization process [6]
The Harmony search algorithm has been successfully applied
to vehicle routing problem [6], hydrologic parameter
calibration [15] and to the Storage space allocation problem
[2]
The HS algorithm includes five steps: parameters
initialization, the harmony memory (HM) initialization, the
new harmony improvisation, the harmony memory update and
the check of termination criterion.
4.1 Parameters initialization
In this step, the optimization problem is specified:
Minimize (or Maximize) f (x); xi Xi, i = 1,2,..., N
Where:
 f(x) is an objective function
 x is the solution vector composed of decision variables
xi
 Xi is the set of possible values for each decision variable
 Xi = {xi (1), xi (2),..., xi (K)} for discrete variables
 N is the number of decision variables
 K is the number of possible value for each discrete
variable
The algorithm parameters are also specified during this step
such as:
 The harmony memory size (HMS) is the number of
solution in the memory
 The harmony memory considering rate (HMCR);
0≤ HMCR ≤1; his typical values range from 0.7 to 0.99
 The pitch adjustment rate (PAR) : 0≤ PAR ≤1; its
selected values range is from 0.1 to 0.5
 Improvisations number.
4.2 Harmony memory initialization
During this step, a harmony memory of size HMS, shown in
equation (9), is randomly generated. Each decision variable
(xi) randomly selects a value from its list (Xi). Then, their
fitness values are calculated.
The harmony memory is initially crammed; a new harmony
vector x’ = (x’1, x’2,.., x’N ) is generated and compared to
existing solutions. It’s kept if it’s better than the worst
harmony.
x' is improvised using the following two rates:
 Harmony memory consideration rate
 Pitch adjustment rate.
The value for each decision variable
is randomly chosen
using a harmony memory consideration rate (HMCR).
The value of
is selected from the pitches previously stored
in HM for this decision variable with a probability HMCR.
While it is chosen from the set of all possible values for the
corresponding decision variable, with a probability (1HMCR).

x i'  x1i , x i2 ,...,x iHMS
x i'  
x i'  X i

w.p HMCR
(10)
w.p (1 - HMCR)
While improvising the new harmony, each value chosen from
HM is examined to determine whether it should be pitchadjusted. This procedure uses the PAR parameter that sets the
rate of adjustment for the pitch chosen from the HM as
follows.
x '  rand(() * bw

x i'   i
x i'
w.p HMCR PAR
w.p HMCR (1 - PAR)
(11)
The value of (1-PAR) sets the rate of doing nothing.
bw: arbitrary distance bandwidth and rand () is a random
number between 0 and 1.
4.4 Harmony memory update
The new solution is stored in the harmony memory if it’s
better than the worst of the existing solutions and it respects
all problem constraints.
Steps (4.3) and (4.4) are repeated while the termination
criterion (maximum number of improvisations) is not reached.
5. EVOLUTION PROCEDURE
In this section, the harmony search algorithm proposed is
detailed. An initial harmony memory of size HMS is created.
The decision variables Ci,t(x, y, z, b), represent the possible
locations for the containers according to the allocated storage
area.
Ci,t(x, y, z, b) used four dimensions structure representation.
These dimensions indicate respectively the container
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International Journal of Computer Applications (0975 – 8887)
Volume 48– No.22, June 2012
coordinates in the plan (X, Y, Z) and the number of the
allocated block.
The figure 4 shows an example of solution representation
6.1 The number of containers type influence
This part studied the influence of the containers type number
since it is an important factor in this problem. The algorithm
is executed for different values of NT and each time the best
fitness values of the first (Fi) and the last iterations (Ff) are
given. Also, the execution time (TExe) is indicated. The
population size was set to 50, the stopping criteria (Niter) to
20, n1 = n2 = n3= 3, Nstock_reg = 4 and Nstock_refrig = 4.
The simulation results are illustrated in table 1.
According to these results, it is clear that higher is the number
of container type important is the execution time and worse is
the fitness value. It’s evident since the complexity of the
problem is directly related to container type number and their
storage constraints.
Fig 4. Example of solution
The initial harmony memory is randomly generated and every
stored solution must respect all problem constraints (equations
(3) to (8)).
After that, a new solution is improvised based on the process
outlined in section 4.3. This step will be repeated until the
termination criterion is satisfied.
Table 1. Container type influence
NT
Nc(T)
Fi
Ff
TExe (s)
1
Nc(1)=10, Nc(2)=10
Nc(1)=10, Nc(2)=10
Nc(3)=8
Nc(1)=10, Nc(2)=10
Nc(3)=8, Nc(4)=8
Nc(1)=10, Nc(2)=10
Nc(3)=8, Nc(4)=8
Nc(5)=15
Nc(1)=10 , Nc(2)=10
Nc(3)=8, Nc(4)=8
Nc(5)=15, Nc(6)=10
2,69
0
3
4,77
0
4,49
28,81
0
7,34
32,84
0
14,21
62,71
0
22 ,54
2
3
Begin creat_solution
Repeat
For j=0 to NBlock -1
For x=0 to n1-1
For y=0 to n2-1
For z=0 to n3-1
Randomly selected a container type (t)
Randomly selected a container i of this
type from ones not already stored
If the constraint of this type is satisfied
Then
Ci,t (x, y, z, b) = 1
Update the container stored list
End
End
End
End
End
Until all containers are stored
End
4
5
6.2 The harmony memory size influence
In order to examine the importance of the harmony memory
size, we fixed the following parameters:
 Nc(T) = 5 (dry, empty, open top, tank, reefer) with
Nc(1)= 20, Nc(2)= 20 Nc(3)=15, Nc(5)= 10,
Nc(6)=20.

Niter = 50

n1 = n2 = n3= 3

Nstock_reg = 3 , Nstock_refrig = 3
Table 2. Population size influence
Fig 5. Solution creation algorithm
6. EXPERIMENTAL RESULTS
HMS
Fi
Ff
T Exe (s)
In this section, experimental results are provided to study the
performance of the proposed approach. This algorithm stops
when the solution doesn’t improve after N iter iterations.
It is assumed that:
 n1, n2 and n3 will be defined by the user,
 The containers type NT, the number of each container type
Nc(T) and the storage blocks number (Nstock_refrig, Nstock_reg)
are defined by the user.
 HMCR= 0.95 and PAR = 0.1.
Departure dates of container are also indicated by the user.
10
31,61
7,81
7,96
20
34,54
6,52
8,64
40
29,34
6,31
10,21
60
27,11
4,78
11,03
80
22,84
4,45
15,02
100
28,19
3,24
18,02
The population size (HMS) is varied. His influence on the
fitness value is presented in the table 2.
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International Journal of Computer Applications (0975 – 8887)
Volume 48– No.22, June 2012
The results indicate that higher is the harmony memory size,
better is the value of the fitness function.
7. COMPARATIVE STUDY
In order to evaluate the results generated by the harmony
search approach, a comparative study with a LIFO (Last In
First Out) algorithm and the genetic algorithm (GA) is
presented.
The LIFO algorithm consists on storing in first time the last
placed container in a stack. This principle is applied in most
port container terminals, where a manual planning based on
experience and rules to assign each container to a certain
storage block.
The Genetic algorithm was proposed to solve the same
problem (SSAP for multiple container type) by Ayachi et al.,
[1]
This GA can be described as follows: Initially, a first
generation is randomly generated. Then, a two-point
crossover operator is performed to two parent selected using
the roulette-wheel method. The mutation operator consists of
permuting two randomly selected containers having the same
type.
Five studied cases are defined by varying the containers
numbers and types, to verify the performance of the three
approaches. Table 3 described these instances.
Table 3. Different studied cases description
Instance N°
NT
Nc(T)
1
2
Nc(1)=50, Nc(3)=15
2
3
3
4
4
5
5
6
Nc(1)=25, Nc(2)=25,
Nc(3)=10
Nc(3)=8, Nc(4)=5,
Nc(5)=7,Nc(6)=15
Nc(2)=14, Nc(3)=8 Nc(4)=5,
Nc(5)=7, Nc(6)=15
Nc(1)=25, Nc(2)=14,
Nc(3)=9, Nc(4)=8, Nc(5)=7,
Nc(6)=12
For each case, the problem is solved 15 times and the mean of
fitness values (F) and execution times are calculated.
In this part, it’s supposed that the population size is set to 30,
Niter to 20, n1, n2 and n3= 3, Nstock_reg to 3 and Nstock_refrig to 2.
The results showed in table 4 indicate that the fitness value
generated by the HS algorithm is largely better for all studied
cases for an execution time tolerant and lower than the
execution time for GA.
Table 4. Comparison between LIFO, GA and HS’s fitness
values and execution time
LIFO
Genetic
Harmony
Instance
Algorithm
algorithm
search
N°
F
TExe
(s)
F
TExe
(s)
F
TExe
(s)
1
3,65
0,5
0
20
0
4,44
2
5,59
2
0
22
0
4,99
3
4,72
4
0
37
0
8,78
4
10,14
4,5
1,29
65
0
10,54
5
19,37
6
3,16
80
1,15
17,97
This can be explained by the fact that the genetic algorithms
evaluate simultaneously several solutions. The GA used
selection, crossover and mutation operators to generate a
better solution. Sometimes, this process is not effective
enough to get optimum solution as they might not effectively
preserve important patterns in chromosomes. [15]
The curve shown in the following figure confirms results
described in the table 4.
Fig 6. Comparison between LIFO, GA and HS’s fitness
values
Harmony search algorithm seems well suited to complex
problem. It generates good results within a tolerable time even
with the diversity types of containers and the appearance of
many storage constraints.
8. CONCLUSION
In this study, a harmony search algorithm is applied to solve
the storage space allocation problem for import containers.
In real world case, there are various types of container such as
refrigerated, open side, empty, dry, open top, tank... Each
container type has storage constraints that must be respected
in the allocation process of the storage areas, which let the
problem more difficult. That is refrigerated containers must be
allocated to the blocks equipped by the power point, tank
containers need to be placed on each others, etc.
Despite this difficult, the proposed approach generated good
results in a reasonable execution time. Experimental study
confirms these and shows the effectiveness of the application
of harmony search in the resolution of this problem.
An important extension of this research would be to formulate
the problem as a dynamic storage space allocation in order to
solve and to make decision in real time.
9. REFERENCES
[1] Ayachi, I., Kammarti, R., Ksouri, M., Borne, P., 2010, A
Genetic algorithm to solve the container storage space
allocation problem, IEEE Trans. International conference
on Computational Intelligence and Vehicular System,
Seoul, South Korea
[2] Ayachi I., Kammarti R., Ksouri, M., Borne, P., 2010,
Harmony search algorithm for the container storage
problem, 8th International Conference of Modeling and
Simulation - MOSIM’10, Tunisia.
[3] Bazzazi, M., Safaei, N., Javadian, N., 2009, A genetic
algorithm to solve the storage space allocation problem
in a container terminal, Computers & Industrial
Engineering 36 (2009), p. 1711–1725.
[4] Chen, P., Fu, Z., Lim, A., Rodrigues, B., 2004, Port yard
storage optimization, IEEE Transactions on Automation
Science and Engineering. Vol. 1, p. 26 – 37.
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